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MATH 2321 Lecture Notes - Lecture 22: Tangent Space, Null Character, Voseo

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sangriamacaw663

13 Feb 2019

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*/=/1,-2,-4) = 0 (2, 1,0) =(4,01)= | li j ik. Tuoi: give a parameterization of the tangent plane to s at point r(1,2) Parametric form: (x,y,z) = 11 (1,2) + ari (1, 2) + bir (1,2). [(x, y, z) = (5,1,2)= a (2,1,0) + 6(4,0,1)): find a standard equation for the tangent plane to s at point in (1,2) n =(2,1,0)*(4,0,1) = (1,-2,-4)= (a, b, c). Txo, yo, zo) = (1, 2) = (5,1,2) a(x-xo) *6(y-yo) +((z-zo) = 0. 1(x-5)-2/4-1)- 4(2-2) = 0 (x - 24-42 +5 = 0 7 i: as a level surface (or a standard eguation); normal vector in = nu 10,0) = 1, (0,0) (0, 1,0) =(-1,0,0). Ixo, yo, zo) = 10(0,0)= (0,0,0) => 0(x-0)+0 (4-0)+1/2-8)=0 z= U, v): describe the surface s as a level surface & sketch. level surface f(x, 4,2)=c (un)=(x, y, z) . x=u2 tv2 ) y eur { delete u&v.

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Exercises and Problems for Section 17.1 18. A ie segment hetween (2, 1,3) and (4.3,2 he plane 5 20. A line perpendicular to the plane --y+7 and through the point (1,1,6) 21. The circle of fabus 2 inthe 'y-plane, centered the on- gin, clockwise. 22. The ciecle of radis 2 parallel to the ry-plane, centered the point (0,0, 1), and traversed countenclockwise when viewed from below 23. The circle of radius 2 in the za-plane, cenered at the ori- gin. 24. The circle of radias 3 parallel to the ry-plane, centered at the point (0,0,2) 25. The circle of redius 3 in the ya-plane, centered a the point (0,0,2) 26. The circle of radius 5 parallel to the ya-plane, centered at the poinn (-1,0,-2) 28. The curve y - 27. The curve zy in the zy-plane zy-plane. in the 29, The curveã¨ã¼ã¼3? in the za-plane. 30.The curve in which the plane 1, 2 cuts the surface -2 z 31. The curve y-4-5' through the point (0,4,4), par. allel to the ry-plane. the line. 32. The elipse of major diameter 5 parallel to the y-axis la Exercises 7-17, find parametric equations for 7. The line in the direction of the vector - and through and minor diameter 2 parallel to the z-axis, centered at the point (0, 1,0). &. The line in the direction of the vector ? +2 33. The ellipse of major diameter 6 along the z-axis and mi- and or diameter 4 along the y-axis, centered at the origin. through the point (3, 0,-4). 9. The line 34Â·The ellipse of major diameter 3 parallel to the z-axis The line in the direction of the vector 53 + 2k and In Exercises 35-42, find a parametric equation for the curve dhrough the point (5,-1,1) segment. (There are many possible answers.) 36. Line fromå¡= (-1,-3) to h-(5,2). 38. Semicircle from (0,0,5) to (0,0,-5) in the yz-plane The line in the direction of the vector 37 -3 +E and 35. Line from (-1,2,-3) to (2,2,2). through the point (1,2,3). 12. The line in the direction of the vector ai +2j-3k and 37. Line from P(1,-3,2 to P (4.1.-3). 13. The line through (-3,-2,1) and (-1,-3, -1) through the point (-3,4, -2) with y 2 0. The line through the points (1,5,2) and (5,0,-1).39. Semicircle from (1,0,0) to (-1,0,0) in the zy-plane 5. The line through the points (2,3, -1) and (5,2,0) with y 2 0. 40. Graph of y VE from (1,1) to (16,4). 41. Are of a circle of radius 5 from P=(0,0) to Q IS The line through (3-2,2)and intersecting the y axis at v 2 (10,0). åºThe line intersecting the z-axis at z = 3 and the z-axis 42. Quarter-ellipse from (4,0 3) to (0 -3,3) in the plane